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Parikh vectors
5APK_1 8JWL_1 1JWV_1 Letter Amino acid
17 25 18 R Arginine
8 2 2 C Cysteine
16 13 8 Q Glutamine
19 15 6 H Histidine
11 14 15 I Isoleucine
5 22 16 D Aspartic acid
23 23 20 E Glutamic acid
6 16 12 P Proline
13 12 8 N Asparagine
7 7 9 M Methionine
7 16 20 T Threonine
9 8 4 Y Tyrosine
15 21 13 V Valine
2 9 4 W Tryptophan
17 37 26 A Alanine
12 39 20 G Glycine
34 36 31 L Leucine
12 10 11 K Lycine
14 8 5 F Phenylalanine
18 18 15 S Serine 

5APK_1|Chains A, B|NUCLEAR RECEPTOR ROR-GAMMA|HOMO SAPIENS (9606)
>8JWL_1|Chains A, B, C, D, E, F, G, H|Aldo/keto reductase|Streptomyces xinghaiensis (1038928)
>1JWV_1|Chain A|BETA-LACTAMASE TEM|Escherichia coli (562)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
5APK , Knot 115 265 0.80 40 168 250 
MHNHNHNHNHNHNGGENLYFQGASLTEIEHLVQSVCKSYRETCQLRLEDLLRQRSNIFSREEVTGYQRKSMWEMWERCAHHLTEAIQYVVEFAKRLSGFMELCQNDQIVLLKAGAMEVVLVRMCRAYNADNRTVFFEGKYGGMELFRALGCSELISSIFDFSHSLSALHFSEDEIALYTALVLINAHRPGLQEKRKVEQLQYNLELAFHHHLCKTHRQSILAKLPPKGKLRSLCSQHVERLQIFQHLHPIVVQAAFPPLYKELFS
8JWL , Knot 150 351 0.83 40 197 333 
MGSSHHHHHHSSGLVPRGSHMEYTQLGRIGLKVSRLVLGTMNFGPTTDEAESHAIMDAALDAGINFFDTANVYGWGENKGRTEEILGSWFAQGGDRRDKVVLATKVYGNMGLDGPAWPNHDKLSALNIRRSVDASLKRLGTDHIDLYQFHHVDRDTPWDEIWQAMDVLVRQGKILYVGSSNFAGWNIAQANETAARHGRLGLVSEQCLYNLCERRAEMEVVPAAREYGLGVIAWSPLHGGLLGGAIRKEQEGGNRRAASGRAADALKDPQQREQIQRYEDLLDKHGLEPGEVALAWLLTRPGVTGPIVGPRTADQLASAVRAAELTLTDEVLTALDEIFPGPGPSPEAFAW
1JWV , Knot 117 263 0.82 40 176 256 
HPETLVKVKDAEDQLGARVGYIELDLNSGKILESFRPEERFPMMSTFKVLLCGAVLSRVDAGQEQLGRRIHYSQNDLVEYSPVTEKHLTDGMTVRELCSAAITMSDNTAANLLLTTIGGPKELTAFLHNMGDHVTRLDRWEPELNEAIPNDERDTTMPAAMATTLRKLLTGELLTLASRQQLIDWMEADKVAGPLLRSALPAGWFIADKSGAAERGSRGIIAALGPDGKPSRIVVIYTTGSQATMDERNRQIAEIGASLIKHW

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(5APK_1)}(2) \setminus P_{f(8JWL_1)}(2)|=67\), \(|P_{f(8JWL_1)}(2) \setminus P_{f(5APK_1)}(2)|=96\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:1000000000000110010101101001001100100000000010100110000011000010100000110110001001001100110110010111010000011110111101111010010010000111010011101101110001100110100010110100001110011111010011100000100100010111000100000001110111010100100001001011001011110111111000110
Pair \(Z_2\) Length of longest common subsequence
5APK_1,8JWL_1 163 4
5APK_1,1JWV_1 170 4
8JWL_1,1JWV_1 139 4

Newick tree

 
[
	5APK_1:87.37,
	[
		8JWL_1:69.5,1JWV_1:69.5
	]:17.87
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{616 }{\log_{20} 616}-\frac{265}{\log_{20}265})=98.6\)
Status Protein1 Protein2 d d1/2
Query variables 5APK_1 8JWL_1 127 108.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]